Question

The intuition behind Hotelling’s rule that prices of nonrenewable or depletable resources should rise at a rate equal to the market interest rate, comes from the fact that ‘when a resource is abundant, then consumption today does not involve an opportunity cost of foregone marginal profit in future, since plenty available for both today and the future. So when resources are traded in a competitive market are abundant, P=MC and marginal profit is 0. As resource becomes increasingly scarce, however, consumption today involves an increasingly high opportunity cost of foregone marginal profit in the future. So as resources become increasingly scarce relative to demand, marginal profit (P-MC) grows. The profit created by user scarcity=marginal user cost (MUC)= price- marginal cost of extraction (P-MEC). a. Suppose there is an unlimited availability of a resource with inverse demand function p=12-0.8q and with marginal extraction cost MEC= 4. Suppose the time horizon is 2 periods. What quantity should be extracted in each period? b. Suppose there is a nonrenewable resource with inverse demand function p= 12-0.8q and with MEC= 4. The resource stock, S is finite and = 16 units. Suppose the time horizon is 2 periods and the discount rate is r= 20%. What quantity should be extracted in each period? (Hint: for scarce nonrenewable resource, the present value of marginal net benefits= p- MEC also known as the MUC should be equal across all periods and present value of marginal net benefits at time t= (Pt- MECt)/ (1+r)t )

Answer 1

a. Suppose there is an with inverse demand function p=12-0.8q and with marginal extraction cost MEC= 4. Suppose the time horizon is 2 periods. What quantity should be extracted in each period?

According to the rule, the marginal benefits should be equal to marginal costs in two periods when there is no discounting

12 - 0.8q1 = 4 and 12 - 0.8q2 = 4

8 = 0.8q1 and 8 = 0.8q2

q1 = 10 and q2 = 10.

This gives q1 = q2 = 10. Since q1 + q2 is infinite, there is no discounting and the problem becomes that of static efficiency.

b. Suppose there is a nonrenewable resource with inverse demand function p= 12-0.8q and with MEC= 4. The resource stock, S is finite and = 16 units. Suppose the time horizon is 2 periods and the discount rate is r= 20%. What quantity should be extracted in each period?

Note that we have a limited availability of a resource q1 + q2 = 16. From the view of dynamic efficiency, the present value of marginal net benefits should be equal across two periods. The presnet value of net marginal benefit for period 1 is P1 - MEC1 = 12 - 0.8q1 - 4 = 8 - 0.8q1. Similarly, the present value of net marginal benefit for period 2 is (P2 - MEC2)/(1 + 20%) = (12 - 0.8q2 - 4)/1.2 = (8 - 0.8q2)/1.2

This gives

8 - 0.8q1 =  (8 - 0.8q2)/1.2

1.2( (8 - 0.8q1) =  (8 - 0.8q2)

9.6 - 0.96q1 = 8 - 0.8q2

1.6 = 0.96q1 - 0.8q2

Use q1 = 16 - q2

1.6 = 0.96*(16 - q2) - 0.8q2

1.6 = 15.36 - 1.76q2

This gives q2 = 7.818 and q1 = 8.182

Hence in period 1, 8.182 units and in period 2, 7.818 units should be extracted.